A category internal to some given (∞,1)(\infty,1)-category 𝒞\mathcal{C} is a simplicial object in an (∞,1)-categoryA:Δop→𝒞A : \Delta^{op} \to \mathcal{C} in 𝒞\mathcal{C}, where the object in degree kk is to be thought of as “the object of kk-tuples of composable morphisms” in AA. This is formalized by requiring the canonical morphisms Ak→A1×A0⋯×A0A1A_k \to A_1 \times_{A_0} \cdots \times_{A_0} A_1 (into the iterated (∞,1)-pullback over the source and target maps) to be an equivalence in𝒞\mathcal{C} (the “Segal condition”).

If 𝒞\mathcal{C} happens to be just a 1-category, then this already makes AA an internal category. Generally, however, 𝒞\mathcal{C} comes with its own notion of homotopy, and one must ask in addition that the notion of equivalence in AA is compatible with that in 𝒞\mathcal{C} (the “completeness condition”).

Motivation and introduction

Before coming to the formal definitions below in Definition, here are some words for the reader looking for introduction and orientation into the general problem of formulating categories internally in homotopy theory.

Whatever exactly the right or desired nature of a category internal to an (∞,1)(\infty,1)-category/homotopy theory is (and we will see that there are some subtleties to beware of and some variants to account for), the bare minimum must be that it consists of

a collectionX0X_0 of objects – but we shouldn’t say set of objects, of course, instead the generic terminology is: a type of objects and so we should write

⊢X0:Type\vdash \; X_0 \colon Type;

for each pairx0,x1:X0x_0, x_1 \colon X_0 of objects, an (x0,x1)(x_0,x_1)-dependent type of morphismsX1(x0,x1)X_1(x_0,x_1) “from x0x_0 to x1x_1,

x0,x1:X0⊢X1(x0,x1)x_0, x_1 \colon X_0 \;\vdash \; X_1(x_0, x_1).

(One might decide to collect these all together to a single type X1≔∑x0,x1:TypeX1(x0,x1)X_1 \coloneqq \underset{x_0,x_1 \colon Type}{\sum} X_1(x_0,x_1), but the theory flows much more naturally if we do keep the dependency on the objects explicit.)

to be thought of as taking a pair(g,f)(g,f) of composable morphisms to their composite morphism g∘fg \circ f, and demand that it satisfies associativity up to homotopy. This approach is explored below in the section H-category types, where it is also discussed that this is not the correct notion of category objects internal to a homotopy-theoretic context.

To get a hint for what the correct formulation should be, it is useful to turn this around and investigate which internal-homotopy-theory structure an ordinarysmall category (internal toSet) gives rise to.

Namely, bare homotopy theory is about groupoids and then n-groupoids and ∞-groupoids, and so it should be relevant that a groupoid is itself a category and that, conversely, every category C∈Cat(Set)C \in Cat(Set) already contains some “homotopy theory”, namely its maximal groupoid, its coreiC:core(C)→Ci_C \colon core(C) \to C. This groupoid is “the homotopy theory of the objects of CC” in the sense that is contains all the information about the objects and their equivalences. Therefore it is natural to regard this as the “type of objects” and declare X0≔core(C)X_0 \coloneqq core(C), regarded as an object of objects in the ambient (∞,1)-categoryGrpd↪\hookrightarrow∞Grpd.

But once we take that perspective, it is clear what the type X1X_1 of morphisms should be: it should be the comma object of the core inclusion with itself: X1≔(iC/iC)X_1 \coloneqq (i_C/i_C).

In traditional category theory a simplicial set is the nerve of a category if and only if it satisfies the Segal conditions. Does the converse already hold here? The above inspection shows that instead of the core inclusion iC:core(C)→Ci_C \colon core(C) \to C we could have started with anyfunctori:K→Ci \colon K \to C and Xn≔i/nX_n \coloneqq i^{/^n} would still defined a simplicial groupoid that satisfies the Segal conditions. So in homotopy theory the Segal conditions, which witness the fact that we formed a nerve by iterated homotopy fiber product, need to be accompanied by one more condition which ensures that we are indeed forming the homotopy fiber product not of any map, but of the core-inclusion (this is often, but somewhat undescrptively, called the “compleness condition”, and more recently also called a univalence condition).

But of course the point of category objects internal ot an (∞,1)(\infty,1)-category is to speak about structures of higher homotopical degree, hence about untruncated simplicial objects in an (∞,1)-category and the appropriate conditions on them to qualify as internal categories. The comprehensive discussion of this definition we turn to in Definition.

The archetypical example here is the case where again H=\mathbf{H} =∞Grpd=:(∞,0)Cat=: (\infty,0)Cat and where then, by induction on nn, the inclusion is ∞Grpd↪(∞,n)Cat\infty Grpd \hookrightarrow (\infty,n)Cat that of ∞\infty-groupoids into (∞,n)-categories. The resulting internal category objects are then externally essentially (n+1)-fold complete Segal spaces, hence (∞,n+1)(\infty,n+1)-categories.

where Δ0op→Δ\Delta_0^{op} \to \Delta is the wide subcategory of the simplex category on the injective maps that moreover send elementary edges to elementary edges (morphisms of linear graphs), and the bottom morphism is the functor that sends a graph object X1→∂0→∂1X0X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 to the object which in degree nn is X1×X0⋯×X0X1⏟nfactors\underbrace{ X_1 \underset{X_0}{\times} \cdots \underset{X_0}{\times} X_1}_{n\; factors}.

Groupoid objects

A groupoid object is a pre-category object, def. 7 whose composition operation according to remark 3 is invertible. This is discussed in more detail at groupoid object in an (∞,1)-category. Here we briefly recall the

Remark

For 𝒞=∞Grpd\mathcal{C} = \infty Grpd, a groupoid objectXX in 𝒞\mathcal{C} is a pre-category object X•∈PreCat(𝒞)↪𝒞ΔopX_\bullet \in PreCat(\mathcal{C}) \hookrightarrow \mathcal{C}^{\Delta^{op}}, def. 7, such that the full inclusion

Remark

We are to think of X([0])X([0]) as the 𝒞−\mathcal{C}-object of objects of a groupoid internal to 𝒞\mathcal{C}, and of X([1])X([1]) as its 𝒞\mathcal{C}-object of morphisms. In terms of this the above condition says two things:

for S∪S′S \cup S' an order-preserving partition (meaning that for all s∈Ss \in S, s′∈S′s' \in S' we have s≤s′s \leq s') it says that X([n])X([n]) may be identified with the object of sequences of lenght nn of composablemorphisms;

for S∪S′S \cup S' not order-preserving, it says that morphisms have inverses. For instance for the partition of [2][2] given by S={0,1}S = \{0,1\} and S′={0,2}S' = \{0,2\} the above says intuitively that diagrams in the internal groupoid of the form

Definition

For X•∈∞GrpdΔopX_\bullet \in \infty Grpd^{\Delta^{op}} a pre-category object, we say that a point f:*→X1f \colon * \to X_1 in the degree-1 object is an equivalence if it is an isomorphism in the category hXh X, def. 4. For n∈ℕn \in \mathbb{N}, n≥1n \geq 1, write

Equiv(Xn)↪Xn
Equiv(X_n) \hookrightarrow X_n

for the 1-monomorphism that includes the full-sub-∞\infty-groupoid on the sequences of equivalences. Write furthermore

Remark

In order to state the completeness condition on pre-category objects that make them be genuine category objects, we need this core-construction exhibits groupoid objects as a coreflective sub-(∞,1)-category of pre-category objects.

Proposition

Let H\mathbf{H} be an (∞,1)-topos. Every groupoid object inH\mathbf{H} is canonically an internal pre-category. Under this inclusion i:Grpd(H)↪PreCat(H)i \colon Grpd(\mathbf{H}) \hookrightarrow PreCat(\mathbf{H}) the groupoid objects form a coreflective sub-(∞,1)-category of that of pre-catgegory objects,

Proof

With def. 5 it is direct to establish the statement for the case that 𝒞≃\mathcal{C} \simeq ∞Grpd, (Lurie, cor. 1.1.11), for instance by using the standard theory of Segal spaces. From this it follows also for the case that H≃PSh∞(𝒟)\mathbf{H} \simeq PSh_\infty(\mathcal{D}) is an (∞,1)-category of (∞,1)-presheaves by arguing objectwise over objects in 𝒟\mathcal{D}. In the general case, H\mathbf{H} is a reflective sub-(∞,1)-category of such, 𝒞↪PSh∞(𝒟)\mathcal{C} \hookrightarrow PSh_\infty(\mathcal{D}). It is then sufficient to show that the core operation on the presheaf ∞\infty-toposes respects these inclusions, so that we have

This means that we need to show that if X•X_\bullet is degreewise in H↪PSh∞(𝒟)\mathbf{H} \hookrightarrow PSh_\infty(\mathcal{D}) and is a pre-category object, then CorePSh(X•)Core_{PSh}(X_\bullet) is degreewise in H\mathbf{H}. By the pre-category condition and since the refletive inclusion is right adjoint and hence preserves the fiber products, for this it is sufficient that CorePsh(X)0Core_{Psh}(X)_0 and CorePSh(X)1Core_{PSh}(X)_1 are in H\mathbf{H}. To complete the proof it is sufficient to show that the first of these is X0X_0 and (again since the inclusion preserves limits) the second is equivalent to the poweringX(K)X(K), where

is the simplicial set obtained from the 3-simplex by collapsing the (0,2)(0,2)-edge and the {1,3}\{1,3\}-edge. Write K0≔{1,2}↪KK^0 \coloneqq \{1,2\} \hookrightarrow K for the image of the (1,2)(1,2)-edge of Δ3\Delta^3. Schematically:

is the space of those 3-simplices in X•X_\bullet for which the {0,1}\{0,1\}-edges is a weak inverse to the {2,3}\{2,3\}-edge. Hence Core(X)(K)→X(K)Core(X)(K) \to X(K) is an equivalence. Moreover K0↪KK^0 \hookrightarrow K is a weak equivalence, and hence so is Core(X)(K)→Core(X)(K0)≃Core(X)1Core(X)(K) \to Core(X)({K^0}) \simeq Core(X)_1, by this proposition at simplicial object in an (∞,1)-category.

Category objects

In an (∞,1)(\infty,1)-topos

Let H\mathbf{H} be an (∞,1)-topos. Then every object of H\mathbf{H} may already be thought of as being an internal groupoid, which facilitates the definition of internal categories. This we discus here. The more general case where the ambient (∞,1)(\infty,1)-category is not necessarily an (∞,1)-topos is discussed further below.

Definition

An internal category in an (∞,1)-toposH\mathbf{H} is an internal pre-category X•∈HΔopX_\bullet \in \mathbf{H}^{\Delta^{op}}, def. 7, such that its coreCore(X)Core(X) is in the image of the inclusion const:H↪Grpd(H)const \colon \mathbf{H} \hookrightarrow Grpd(\mathbf{H}) of prop. 1.

Remark

A groupoid object, def. 3, is always a pre-category object, but is a category object only and precisely if it is in the image of the constant inclusion Const:H→Grpd(H)Const \colon \mathbf{H} \to Grpd(\mathbf{H}).

In a sense these are the genuine groupoid objects, while the others are groupoid objects equipped with an atlas (…).

For internalizing in an (∞,1)(\infty,1)-category 𝒞\mathcal{C} which is not an (∞,1)-topos, we need to specify what the constant groupoid objects in 𝒞\mathcal{C} are supposed to be. This is the topic of

works essentially as before in an (∞,1)(\infty,1)-topos. The key point is that the ambient (∞,1)-toposH\mathbf{H} serves itself naturally as the collection of groupoid objects inside its (∞,1)-category of inernal categories and so this yields a natural notion of

Example

The identity H≃𝒞\mathbf{H} \simeq \mathcal{C} is a choice of internal groupoids in H\mathbf{H}, by the (∞,1)-Giraud theorem. For this choice the following theory of category objects in 𝒞\mathcal{C} relative to H\mathbf{H} reduces to that of category objects in H\mathbf{H}, as discussed above.

For the discussion of (∞,n)-categories, the central property of such choices of internal groupoids, def. 6 is that they behave well with forming internal categories, this is cor. 2 below.

Write PreCatH(𝒞)PreCat_{\mathbf{H}}(\mathcal{C}) for the (∞,1)(\infty,1)-category of internal pre-categories in 𝒞\mathcal{C} relative to H\mathbf{H}, the full sub-(∞,1)-category of the simplicial objects on the internal precategories.

Proof

The left morphism has a right adjoint by prop. 2. The right adjoint of the right functor is implied by the first of the axioms on the choice of groupoids Disc:H↪𝒞Disc \colon \mathbf{H} \hookrightarrow \mathcal{C}, by which DiscDisc has a right adjoint Γ\Gamma and is itself a right adjoint. This means that their prolongations to simplicial objects both preserve pre-category objects and hence induce an adjunction of pre-category objects. Moreover, since DiscDisc is in addition fully faithful, Γ\Gamma takes objects in the inclusion back to themselves, and hence this preserves also (H↪𝒞)(\mathbf{H}\hookrightarrow \mathcal{C})-relative precategory objects.

Definition

An internal pre-category X•𝒞ΔopX_\bullet \mathcal{C}^{\Delta^{op}}, def. 7, is called an internal category if its H\mathbf{H}-Core, def. 3, is an essentially constant groupoid object, hence if

Corollary

Remark

So far we have only ever used the first axiom in def. 6. We now describe the reflector PreCatH(𝒞)→CatH(𝒞)PreCat_{\mathbf{H}}(\mathcal{C}) \to Cat_{\mathbf{H}}(\mathcal{C}) in more detail, and for that we use the other two axioms.

The corresponding reflector is “Segal completion”. We now describe this in more detail.

Definition

For X•,Y•∈PreCatℋ(𝒞)X_\bullet, Y_\bullet \in PreCat_{\mathcal{H}}(\mathcal{C}), a morphismf•:X•→Y•f_\bullet \colon X_\bullet \to Y_\bullet of pre-category objects (hence of the underlying simplicial objects) is a categorical equivalence if

Remark

In particular, by reflectivity, this means that a morphism f•:X•→Y•f_\bullet \colon X_\bullet \to Y_\bullet in CatH(𝒞)Cat_\mathbf{H}(\mathcal{C}) is an equivalence (hence an equivalence in 𝒞Δop\mathcal{C}^{\Delta^{op}}) precisely if it is a categorical equivalence.

Iterated internalization – Internal nn-categories

A central point of the formulation of internal category objects is that it can be iterated to yields categories objects internal to category objects … internal to an (∞,1)(\infty,1)-topos.

Proposition

Let H↪𝒞\mathbf{H} \hookrightarrow \mathcal{C} be a choice of internal groupoids, def. 6. Then also the constant inclusion

H↪CatH(𝒞)
\mathbf{H} \hookrightarrow Cat_{\mathbf{H}}(\mathcal{C})

into the (∞,1)(\infty,1)-category of the corresponding category objects is a choice of internal groupoids.

is fully faithful, since Δop\Delta^{op} is a contractible (∞,1)(\infty,1)-category . The first inclusion preserves limits and colimits since in presheaf categories these are computed objectwise, similarly for the second, using the condition that already Disc:H→𝒞Disc \colon \mathbf{H} \to \mathcal{C} preserves limits and colimits. Moreover, this inclusion clearly factors through CatH(𝒞)↪𝒞ΔopCat_{\mathbf{H}}(\mathcal{C}) \hookrightarrow \mathcal{C}^{\Delta^{op}}, by def. 7, and since that is also fully faithful, also H→CatH(𝒞)\mathbf{H} \to Cat_{\mathbf{H}}(\mathcal{C}) preserves limits and colimits.

Enriched categories as internal categories

The 1-categorical notion of enriched category is similar to that of internal category, a main difference being that an internal category has an object of objects in the ambient category 𝒞\mathcal{C}, whereas an enriched category has a set/class of objects. If, however, 𝒞\mathcal{C} is equipped with a notion of discrete objects, thought of as the inclusion of sets into 𝒞\mathcal{C}, then 𝒞\mathcal{C}-enriched categories may be thought of as those categories internal to 𝒞\mathcal{C} such that their object of objects is discrete.

Accordingly, if for 𝒞\mathcal{C} a presentable (∞,1)-category equipped with a choice of internal groupoidsH↪𝒞\mathbf{H} \hookrightarrow \mathcal{C}, def. 6, for H≃\mathbf{H} \simeq ∞Grpd, then an A∈CatH(𝒞)A \in Cat_{\mathbf{H}}(\mathcal{C}) by definition has a bare (external / discrete) ∞\infty-groupoid “of objects”. The underlying simplicial object is thus more like a Segal category object (though still different from that).

In (Lurie, def. 1.3.3) such an choice of internal groupoids ∞Grpd↪𝒞\infty Grpd \hookrightarrow \mathcal{C} is called an “absolute distributor”.

an object Δop→C\Delta^{op} \to C is fibrant precisely if it is fibrant in [Δop,C]proj/inj/Reedy[\Delta^{op}, C]_{proj/inj/Reedy} and if the corresponding simplicial object Δop→C∘\Delta^{op}\to C^\circ in the (∞,1)-categorypresented by CC is an internal category.

However, at the time of this writing little is known for how to speak of non-finitediagrams fully internally. To appreciate the issue, notice that the “internal” formulation of categories by simplicial objects in an (∞,1)-category as above is in fact not fully internal to the ambient (∞,1)(\infty,1)-category 𝒞\mathcal{C}, but assumes that it is known what externally an (∞,1)-functorΔop→𝒞\Delta^{op} \to \mathcal{C} is. When we speak in homotopy type theory this is not an option and we have to genuinely stick to internal reasoning.

Nevertheless, we can speak fully internally of (∞,n)(\infty,n)-categories for “low nn” by making use of the following observations.

If X0X_0 is n-truncated (h-leveln+2n+2) so that we are dealing with an internal category that is externally an (n,1)-category, then a complete semi-Segal object X•X_\bullet should indeed already be determined by its simplicial skeletonX0≤•≤n+2X_{0 \leq \bullet \leq n+2}.

H-Category types

It may be instructive to begin by discussing a notion of internal category in homotopy theory which is not an category object in an (∞,1)(\infty,1)-category and which does not interpret as an (∞,1)-category, but which serves to illustrate a subtlety in the correct definition. In fact, it is an internal formulation of the H-category that underlies in particular any genuine precategory object, by def. 4 above.

The point here is that these homotopies are only required to exist, but are not specified and are not part of the data of an H-monoid.

One also speaks of an H-group for an H-monoid for which the product operation similarly has inverses up to unspecified homotopies. Hence it makes sense to consider the following many-object version (“oidification”) of this classical concept:

One observes that those that areise this way carry much more structure than just a composition and unit up to unspecified homotopy: we can instead make an explicit choice of such homotopies by choosing ways to reparameterize the interval and, crucially, any two such choices are themselves related by a choice of higher order homotopy, and so ever on. Such a structure is called a strong homotopy monoid structure with strong homotopy associativity (as opposed to just bare homotopy associativity as in an H-monoid). Later this was abbreviated to A-∞ structure.

The classical result by Jim Stasheff answered the question by saying that: